Introduction

Abstract

Among various approaches used for the calculation of the stress—strain state in heterogeneous materials the dominant position belongs to numerical and asymptotic methods. The finite element method is one of the most universal numerical procedures. As a rule, it can satisfy the needs of practical engineering, especially concerning the calculations relating to complicated structures like aircraft or ship hulls. However, great developments in numerical methods do not exclude the need for analytical solutions. Moreover, the optimal numerical approaches have to take into account information about the analytical essence of the problem. In this connection we also ought to point out that analytical methods always impose a proper style on the researcher’s thought processes. The fast development of numerical mathematics directly depends on fundamental analytical investigations. Therefore, only harmonious common development of analytical and numerical approaches can provide progress in the theory of materials and structures. Both methods in certain sense complement each other. Finally, it is very important to note that direct applications of any numerical technique can be efficient if the scale of heterogeneity is of the order of the typical outer size of the structure. It is true that the fast development of both hard- and software essentially widens the range of efficient applications of numerical procedures. However, a large number of inclusions or reinforcing elements, which is the case for numerous applications in composites, nonhomogeneous plates and shells, makes direct calculations impossible.